The RO(G)-graded equivariant ordinary cohomology of complex projective spaces with linear ℤ/p actions
The RO(G)-graded equivariant ordinary cohomology of complex projective spaces with linear ℤ/p actions
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具有线性 ℤ/p 作用的复射影空间的 RO(G) 分级等变常上同调
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发表时间：
1988
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通讯作者：
L. Lewis
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作者：
L. Lewis
引言。如果X是仅含偶数维胞元的CW复形，且R是环，则根据胞上同调理论的一个初等结果，X的具有R系数的普通上同调H*(X；R)是自由的7/分次R-模。由于这一结果在射影空间或Grassmannians等行为良好的复杂流形的研究中非常有用，因此能够将其推广到等变普通上同调将是很好的。这一结果确实在以下意义上进行了推广。设G是有限群，X是G-CW复形(在[MAT，LMSM]意义下)，R是环值自变系数系Jill。则具有R系数的X的G-等变普通Bredon上同调H*(X；R)可视为一个系数系。如果X的胞元都是偶数维的，则H*(X；R)在适用于系数系的意义上是R上的自由模。不幸的是，这个定理不适用于复射影空间或具有任何合理的非平凡G-作用的复Grassmannians空间，因为这些空间不具有正确的G-CW结构。事实上，如果G是~/p，对任意素数p，且r~f是非平凡的不可约复G-表示，则该定理不适用于S~，即r1的一点紧化.而且，S n的2~-分次Bredon上同调在Burnside环系数系中的系数显然不是在系数系上自由的.
INTRODUCTION. If X is a CW complex with cells only in even dimensions and R is a ring, then, by an elementary result in cellular cohomology theory, the ordinary eohomology H*(X;R) of X with R coefficients is a free, 7/-graded R-module. Since this result is quite useful in the study of well-behaved complex manifolds like projective spaces or Grassmannians, it would be nice to be able to generalize it to equivariant ordinary eohomology. The result does generalize in the following sense. Let G be a finite group, X be a G-CW complex (in the sense of [MAT, LMSM]), and R be a ring-valued eontravariant coefficient system JILL]. Then the G-equivariant ordinary Bredon cohomology H*(X; R) of X with R coefficients may be regarded as a coefficient system. If the cells of X are all even dimensional, then H*(X;R) is a free module over R in the sense appropriate to coefficient systems. Unfortunately, this theorem does not apply to complex projective spaces or complex Grassmannians with any reasonable nontrivial G-action because these spaces do not have the right kind of G-CW structure. In fact, if G is ~/p, for any prime p, and r / is a nontrivial irreducible complex G-representation, then the theorem does not apply to S ~, the one-point compactification of r 1. Moreover, the 2~-graded Bredon cohomology of S n with coefficients in the Burnside ring coefficient system is quite obviously not free over the coefficient system.